Eigenvector
An eigenvector of a square matrix A is a non-zero vector v that, when multiplied by A, yields a scalar multiple of itself: Av = λv, where λ is the corresponding eigenvalue. Geometrically, eigenvectors are the special directions in space that a linear transformation leaves invariant — only stretching or compressing along those directions without rotation. Eigenvectors are the foundation of matrix diagonalisation, principal component analysis, Google's PageRank algorithm, and the description of quantum states in physics.
Key Formula
A * v = lambda * v, where v is non-zero
LaTeX: A\mathbf{v} = \lambda \mathbf{v}, \quad \mathbf{v} \neq \mathbf{0}
| Symbol | Meaning | Unit |
|---|---|---|
| A | Square n×n matrix | — |
| \mathbf{v} | Eigenvector (non-zero vector) | — |
| \lambda | Eigenvalue corresponding to v | — |
Worked Example
Problem
Find the eigenvectors of A = [[4, 1], [2, 3]] corresponding to λ₁ = 5.
Solution
Step 1: Solve (A − 5I)v = 0. Step 2: A − 5I = [[−1, 1], [2, −2]]. Step 3: Row reduce: [[−1, 1], [0, 0]] (R2 + 2R1 → R2). Step 4: From −v₁ + v₂ = 0 → v₁ = v₂. Step 5: Free variable v₂ = t; eigenvector is v = t(1, 1) for any t ≠ 0.
Answer
Eigenvector for λ = 5 is any scalar multiple of (1, 1), e.g. v = (1, 1).
Eigenvector Properties and Interpretations
| Property | Statement | Significance |
|---|---|---|
| Non-uniqueness | cv is also an eigenvector for any c ≠ 0 | Eigenvectors form a subspace (eigenspace) |
| Orthogonality (symmetric) | Distinct eigenvalues → orthogonal eigenvectors | Basis of symmetric transformations |
| Diagonalisation | A = PDP⁻¹ if eigenvectors span ℝⁿ | Simplifies matrix exponentiation |
| Zero vector excluded | v ≠ 0 by definition | Trivial solution excluded to be meaningful |
| Geometric multiplicity | Dimension of eigenspace | Number of independent eigenvectors for λ |
Interactive Tools
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Related Terms
Eigenvalue
An eigenvalue of a square matrix A is a scalar λ such that there exists a non-zero vector v (its eigenvector) satisfying Av = λv, meaning the matrix only scales the vector without changing its direction. Eigenvalues are found by solving the characteristic equation det(A − λI) = 0, and they encode fundamental properties of the linear transformation including stability, principal stresses, and oscillation frequencies. Applications span quantum mechanics (energy levels), structural engineering (natural frequencies), principal component analysis (data variance), and PageRank (web link importance).
Matrix (Linear Algebra)
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns, where an m×n matrix has m rows and n columns. Matrices represent linear transformations between vector spaces, encode systems of linear equations, and serve as the primary computational tool in linear algebra. They are indispensable across engineering, physics, computer science, statistics, and machine learning, where they model data, rotations, reflections, and the weights of neural networks.
Orthogonality
Two vectors are orthogonal if their dot product is zero, which generalises the geometric notion of perpendicularity to any vector space equipped with an inner product. A set of vectors is orthogonal if every pair of distinct vectors in the set is orthogonal, and if additionally every vector has unit length, the set is called orthonormal. Orthogonality is fundamental in Fourier series (decomposing signals into orthogonal frequency components), least-squares regression (projecting data onto orthogonal subspaces), quantum mechanics (orthogonal states are mutually exclusive), and numerical linear algebra (orthogonal transformations preserve lengths and angles).
From German "eigen" (own, characteristic) + "Vektor" (vector, from Latin "vector" — carrier). Coined in the same tradition as "eigenvalue" by David Hilbert and colleagues in early 20th-century German mathematics.